• Drawing conclusions based on data while accounting for random variation including sampling and observational errors.

• Goal: Make inference about the state of the world using data.

• Most statisitics is taught as a recipe.

• If your data are X then do Y.

• Where is the creativity? Science?

• Where is the science?

• Don't we know something about the world other than our data is X?

• How do we add this knowledge into our modeling?

• Probability model.

  • Model encodes our understanding of the scientific process of interest.

  • Model accounts for as much uncertainty as possible.

  • Model results in a probability distribution.

• Update model with data.

  • Use the model to generate parameter estimates given data.

• Criticize the model

  • Does the model fit the data well?

  • Do the predictions make sense?

  • Are there subsets of the data that don't fit the model well?

• Make inference using the model.

  • If the model fits the data, use the model fit for prediction or inference.

• Start with probability distributions:

  • Data \(\mathbf{y}\).

  ---

  • Parameters \(\boldsymbol{\theta}\).

  ---

  • \([\mathbf{y}]\) is the probability distribution of \(\mathbf{y}\).

  ---

  • \([\mathbf{y} | \boldsymbol{\theta}]\) is the conditional probability distribution of \(\mathbf{y}\) given parameters \(\boldsymbol{\theta}\).

\[ \begin{align*} \left[y_i | \boldsymbol{\theta} \right] & \sim \operatorname{N}(X_i \beta, \sigma^2) \\ \boldsymbol{\theta} & = (\beta, \sigma^2) \end{align*} \]

• Hierarchical model:

  • A model built in components.

  • Each component represents a different statistical goal.

• Break the model into components:

  • Data Model.

  • Process Model.

  • Prior Model.

• Combined, the data model, the process model, and the prior model define a posterior distribution.

\[ {\huge \begin{align*} \color{red}{[\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{z}]} \end{align*} } \]

• Describes how the data are collected and observed.
  • Account for measurement process and uncertainty.
  • Model the data in the manner in which they were collected.

• Data \(\mathbf{y}\).
  • Noisy data.
  • Inexpensive data.
  • Not what you want to make inference on but close.

\[ {\huge \begin{align*} \color{red}{[\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{z}]} \end{align*} } \]

• Latent variables \(\mathbf{z}\).
  • Think of \(\mathbf{z}\) as the ideal data.
  • No measurement error - the exact quantity you want to observe but can't.

• Data model parameters \(\boldsymbol{\theta}_D\).

\[ {\huge \begin{align*} \color{red}{[\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{z}]} \end{align*} } \]

• Age of minerals:

  • \(\mathbf{y}\) is the radio-date estimate.

  • \(\mathbf{z}\) is the true mineral age.

  • \(\theta_D\) is the radio-date standard error.

  • The probability distribution is determined by the measurement process.

\[ {\huge \begin{align*} \color{red}{[\mathbf{y} | \boldsymbol{\theta}_D, \mathbf{z}]} \end{align*} } \]

• Reconstructing climate from tree rings

  • \(\mathbf{y}\) is the tree ring width increment.

  • \(\mathbf{z}\) is the true, unobserved climate variable.

  • \(\boldsymbol{\theta}_D\) models the relationship between climate, stand dynamics, individual heterogeneity, tree age, (etc.) and tree ring width.

  • The probability distribution is determined by tree physiology, measurement uncertainty, etc.

\[ {\huge \begin{align*} \color{blue}{[\mathbf{z} | \boldsymbol{\theta}_P]} \end{align*} } \]

• Where the science happens!

• Latent process \(\mathbf{z}\) is modeled given data \(\mathbf{y}\).
  • Can be dynamic in space and/or time

• Process parameters \(\boldsymbol{\theta}_P\).

• Virtually all interesting scientific questions can be made with inference about \(\mathbf{z}\)

\[ {\huge \begin{align*} \color{blue}{[\mathbf{z} | \boldsymbol{\theta}_P]} \end{align*} } \]

• Sediment Mixing:
  • Model different mineral creation events.
  • Model mixing of sediments over time.
  • \(\mathbf{z}\) includes the true unobserved mineral age as well as the discrete mineral creation event.
  • \(\boldsymbol{\theta}_P\) includes the duration of the minearl creation event, the number of mineral creation events, and the relative mixing of rock to produce a sediment.

\[ {\huge \begin{align*} \color{blue}{[\mathbf{z} | \boldsymbol{\theta}_P]} \end{align*} } \]

• Reconstructing climate with tree rings

  • Trees of the same species share a similar response to climate.

  • Climate variables at sites nearby in location are closer to each other than sites far apart, on average.

  • Climate variables seperated by short periods of time are more similar than climate variables over long periods of time.

  • \(\mathbf{z}\) is the value of the unobserved climate variables.

  • \(\boldsymbol{\theta}_P\) are the species-specific growth responses and the correlation of climate across time and space.

• Probability distributions define "reasonable" ranges for parameters.

• Prior models are useful for a variety of problems:
  • Choosing important variables.
  • Preventing overfitting (regularization).
  • "Pooling" estimates across categories.

\[ {\huge \begin{align*} \color{cyan}{[\mathbf{z}, \boldsymbol{\theta}_D, \boldsymbol{\theta}_P | \mathbf{y}]} \end{align*} } \]

• Probability distribution over all unknowns in the model.

• Inference is made using the posterior distribution.

• Because the posterior distribution is a probability distribution, uncertainty is easy to calculate.

• Climate change is well understood globally.

• Climate change is less well understood locally.

• Need for spatially explicit reconstructions of climate variables.

• Problem: data sources are messy and noisy.

• Vegetation composition and structure change from ice age to current period.

• Using change in temperature to predict future vegetation change.

• Classify compositional and structural change.

• Data model: Multi-logit distribution for ordered categories of observed change.

• Process model: Assumes increasing temperature results in smooth changes of composition and struction.

• Prior model: Not used.

Sharman and Johnstone (2017). Sediment unmixing using detrital geochronology. Earth and Palenetary Science Letters.

• Estimate proportion of each parent in a daughter.

• Reconstruct unobserved parent distributions given daughters.

• Dating uncertainty.

• \(y_{ib}\): age measurement on mineral \(i=1, \ldots, N_b\) for parent \(b = 1, \ldots, B\).

• \(y_{id}\): age measurement for mineral \(i=1, \ldots, N\) of daughter \(d\).

• Measurement error reported as standard deviation \(\sigma_{ib}\) (\(\sigma_{id}\)).

\[ \begin{align*} y_{ib} & \sim \operatorname{N}(z_{ib}, \sigma^2_{ib}). \\ y_{id} & \sim \operatorname{N}(z_{id}, \sigma^2_{id}). \end{align*} \]

Assumptions:

• An unknown number of mineral creation events that are relatively discrete in geologic time.

• Each parent is an mixture of minerals from creation events.

• Each daughter is a mixture of parents.

• For each parent \(b\), the proportion of creation events is \(\mathbf{p}_b\) where \(p_{bk} > 0\) and \(\sum_{k=1}^\infty p_{bk} = 1\).

• Most of the \(p_{bk}\)s are 0 (reasonable in real world).

• Unkown number of mineral creation events.

\[ \begin{align*} {z}_{ib} \sim \sum_{k=1}^\infty p_{bk} \operatorname{N}(\mu_k, \sigma^2_k). \end{align*} \]

• Assigns observations to clusters.

• Model for \(\mathbf{p}_b\).

• Number of clusters increases with number of observations.

• Daughter is a mixture of parents.

\[ \begin{align*} z_{id} & \sim \sum_{b=1}^B \phi_b \sum_{k=1}^K p_{bk} \operatorname{N}(\mu_k, \sigma^2_k). \end{align*} \]

• \(\phi_b\) is the proportion of daughter sediments from parent \(b\).

\[ \begin{align*} \phi_1 = 0.200 \quad\quad\,\,\, \phi_2 = 0.532 \quad\quad\,\,\,\,\, \phi_3 = 0.268 \,\,\quad\quad\quad \mbox{Daughter} \end{align*} \]

\[ \begin{align*} y_{id} & \sim \operatorname{N}(z_{id}, \sigma^2_{id}). \end{align*} \]

\[ \begin{align*} z_{id} & \sim \sum_{b=1}^B \phi_{db} \sum_{k=1}^K p_{bk} \operatorname{N}(\mu_k, \sigma^2_k). \end{align*} \]

• Dirichlet process model can be used to describe sediment mixing processes.

• Estimation of mixing and reconstruction with uncertainty.

• Can ask questions like: what is probability at least 50% of sediment from daughter \(d\) is from parent \(b\):

\[ \begin{align*} \sum_{k=1}^K I\{ \phi_b^{(k)} > 0.5 \}. \end{align*} \]

• Account for spatial correlation among daughters.

• Account for temporal correlation within a sediment core.

• Many ecological and physical processes respond to climate over different time scales.
  • Tree rings, corals, forest landscapes, ice rings, lake levels, etc.

• These processes are called climate proxies.
  • They are proxy measurements for unobserved climate.
  • Noisy and messy.
  • Respond to a wide variety of non-climatic signals.